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A176999 An encoding of the Collatz iteration of n. 3
1, 1111010, 11, 11110, 11110101, 1111011101101010, 111, 1111011101101010110, 111101, 11110111011010, 111101011, 111101110, 11110111011010101, 11110111110101010, 1111, 111101110110, 11110111011010101101, 11110111011010111010, 1111011, 1111110, 111101110110101 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Working from right to left, the sequence of 0's and 1's in a(n) encode, respectively, the sequence of 3x+1 and x/2 steps in the Collatz iteration of n. This is reverse one's complement of Garner's parity vector. Criswell mentions this encoding.

The length of a(n) is A006577(n). The number of 1's in a(n) is A006666(n). The number of 0's in a(n) is A006667(n). The number of terms having length k is A005186(k).

LINKS

T. D. Noe, Table of n, a(n) for n = 2..10000

Evans A. Criswell, The Collatz Problem (3x+1)

Lynn E. Garner, On heights in the Collatz 3n+1 problem, Discrete Math, 55 (1985), 57-64.

EXAMPLE

a(5)=11110 because the Collatz iteration for 5 is a 3x+1 step (0) followed by 4 x/2 steps (four 1's).

MATHEMATICA

encode[n_]:=Module[{m=n, p, lst={}}, While[m>1, p=Mod[m, 2]; AppendTo[lst, 1-p]; If[p==0, m=m/2, m=3m+1]]; FromDigits[Reverse[lst]]]; Table[encode[n], {n, 2, 26}]

CROSSREFS

Sequence in context: A235221 A060087 A229783 * A035613 A038449 A262498

Adjacent sequences:  A176996 A176997 A176998 * A177000 A177001 A177002

KEYWORD

nonn

AUTHOR

T. D. Noe, Apr 30 2010

STATUS

approved

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Last modified January 22 21:15 EST 2022. Contains 350504 sequences. (Running on oeis4.)